Focal Boundary Value Problems for Differential and Difference Equations

301 Pages·1998·12.96 MB·English

by  Ravi P. Agarwal (auth.)

Focal Boundary Value Problems
for Differential and Difference Equations
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centrefor Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 436
Focal Boundary Value
Problems for
Differential and
Difference Equations
by
Ravi P. Agarwal
Department 0/ Mathematics,
National University 0/ Singapore,
Singapore
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5005-2 ISBN 978-94-017-1568-3 (eBook)
DOI 10.1007/978-94-017-1568-3
Printed on acid-free paper
All Rights Reserved
@1998 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1998
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Dedicated to
Robert P Gilbert
on his 65th birthday
Contents
Preface ix
Chapter 1
Continuous Problems
1.1. Introduction 1
1.2. Abel-Gontscharoff Interpolation 4
1.3. Solution of Linear Problems 14
1.4. Existence and Uniqueness 16
1.5. Picard's and Approximate Picard's Methods 21
1.6. Quasilinearization and Approximate Quasilinearization 30
1.7. Integro-Differential Equations 42
1.8. Delay-Differential Equations 47
1.9. Necessary and Sufficient Conditions for Right Disfocality 56
1.10. Tests for Right and Eventual Disfocalities 67
1.11. Green's Functions 75
1.12. Monotone Convergence 87
1.13. Uniqueness Implies Uniqueness 92
1.14. Uniqueness Implies Existence 99
1.15. Continuous Dependence and Differentiation
with Respect to Boundary Values 108
1.16. Right Disfocality Implies Right Disfocality 113
1.17. Right Disfocality Implies Existence 115
1.18. Differential Inequalities Imply Existence 119
1.19. Infinite Interval Problems 122
1.20. Best Possible Results: Control Theory Methods 127
1.21. Converse Theorems 137
1.22. Focal Subfunctions 148
1.23. Generalized Problem I 156
1.24. Generalized Problem 11 160
1.25. A Singular Problem 169
1.26. A Problem with Impulse Effects 177
Comments and Remarks 183
References 184
viii
Chapter 2
Discrete Problems
2.1. Introduction 192
2.2. Discrete Abel-Gontscharoff Interpolation 195
2.3. Existence and Uniqueness 206
2.4. Picard's and Approximate Picard's Methods 210
2.5. Quasilinearization and Approximate Quasilinearization 214
2.6. Necessary and Sufficient Conditions for Right Disfocality 217
2.7. Tests for Right and Eventual Disfocalities 228
2.8. Green's Functions 236
2.9. Monotone Convergence 252
2.10. Continuous Dependence and Differentiation
with Respect to Initial and Boundary Values 254
2.11. Differences with Respect to Boundary Points 263
2.12. Uniqueness Implies Existence 267
2.13. Generalized Problems 277
Comments and Remarks 282
References 283
Index 288
PREFACE
The last fifty years have witnessed several monographs and hundreds of
research articles on the theory, constructive methods and wide spectrum of
applications of boundary value problems for ordinary differential equations.
In this vast field of research, the conjugate (Hermite) and the right focal point
(Abei) types of problems have received the maximum attention. This is largely
due to the fact that these types of problems are basic, in the sense that the
methods employed in their study are easily extendable to other types of prob
lems. Moreover, the conjugate and the right focal point types of boundary
value problems occur frequently in real world problems. In the monograph
Boundary Value Problems for Higher Order Differential Equations published
in 1986, we addressed the theory of conjugate boundary value problems. At
that time the results on right focal point problems were scarce; however, in
the last ten years extensive research has been done. In Chapter 1 of the mono
graph we offer up-to-date information of this newly developed theory of right
focal point boundary value problems.
Until twenty years ago Difference Equations were considered as the dis
cretizations of the differential equations. Further, it was tacitly taken for
granted that the theories of difference and differential equations are parallel.
However, striking diversities and wide applications reported in the last two
decades have made difference equations one of the major areas of research. In
the monographs Difference Equations and Inequalities and Advanced Topics
in Difference equations published in 1992 and 1997, we have devoted substan
tial parts to the study of boundary value problems. But, before these books
were completed many questions for the discrete focal point problems were un
resolved. In Chapter 2 of the monograph we present the complete theory of
discrete focal point boundary value problems.
We would like to take this opportunity to thank J. Henderson, V. Laksh
mikantham, Doanl O'Regan, P.Y.H. Pang, Q. Sheng and P.J.Y. Wong for their
help in writing this monograph. Special thanks go to Sadhna for her constant
encouragement and support.
Ravi P Agarwal
CHAPTER 1
Continuous Problems
1.1. Introduction
In this chapter, we shall mainly consider the following n-th order ordinary
differential equation
x(n) = f(t,x,x', ... ,x(q»), 0::;q::;n-1, butfixed (1.1.1)
together with the right foeal point (Abel-Gontseharoff [15,40,67,170]) boundary
eonditions
(1.1.2)
where -00 < a ::; a1 ::; a2 ::; ... ::; an ::; b < 00. Throughout, unless otherwise
stated, it will be assumed that the function f in (1.1.1) is continuous at least
in the interior of the domain of its definition. Boundary conditions (1.1.2) are,
in asense, equivalent to the (k1, ... , kr) right foeal point eonditions
j
X(;)(aJ - ) -- A-t,-J' Sj-1 ::; i::; Sj -1, So = 0, Sj = Lk;,
;=1
LT (1.1.3)
k; 2: 1, 1 ::; j ::; r(2: 2), k; = n,
;=1
Special attention will be given to the two-point (k1, k2) right foeal eonditions
x(i)(al) O::;i::;a (a+1=k
1)
xci) (a2) a + 1 ::; i ::; n - 1 (n - a - 1 = k2) (1.1.4)
a ::; a1 < a2 ::; b.
=
For convenience, sometimes we shall also assume that a k - 1, i.e.,
k1 = k, and refer to (1.1.4) as the two-point (k, n - k) right foeal eonditions.
Often, we will refer to results concerning solutions of (1.1.1) which satisfy
boundary conditions of the form
R. P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations
© Springer Science+Business Media Dordrecht 1998
2 Continuous Problems
o
~ i ~ kj - 1, 1 ~ j ~ r,
r
L
ki ~ 1, 1 ~ j ~ r (~ 2), ki = n, (1.1.5)
i=l
Such a problem is called an r-point eonjugate boundary value problem for
(1.1.1), and it has been a subject matter of intense study, see [8] for more
detailed information.
For [al, ar] C (a, b), if each (kl ,···, kr) right focal point boundary value
problem for (1.1.1) has at most one solution on (a, b), then (1.1.1) is said to
be (k k right disfoeal on (a, b). Further, if (1.1.1) is (k k right
l,···, r) l ,···, r )
disfocal on (a, b), for all 2 ~ r ~ n, then (1.1.1) is said to be right disfoeal
on (a, b). Similarly, for [al, ar] c (a, b), 2 ~ r ~ n, if each r-point conjugate
boundary value problem for (1.1.1) has at most one solution on (a, b), then
(1.1.1) is said to be diseonjugate on (a, b).
If al ~ a2 ~ ... ~ an, then (1.1.2) are called left focal point boundary
conditions. Similarly, if al > a2 > ... > a then (1.1.3) are called (k k
r l,···, r )
left focal condi tions. The (kl,···, kr) left disfocali ty and the left disfocali ty
on (a, b) are defined analogously.
The plan of this chapter is as follows: In Seetion 1.2 we provide two dif
ferent representations of the Abel-Gontscharoff interpolating polynomial, and
establish best possible error inequalities for its derivatives. In Section 1.3 we
formulate a very simple constructive method for the approximate solutions of
linear right focal point boundary value problems. The inequalities offered in
Section 1.2 are of immense value in the study of right focal point boundary
value problems (1.1.1), (1.1.2). Indeed, we shall use these inequalities in Sec
tions 1.4-1.6 to provide necessary and sufficient conditions for the existence
and uniqueness of solutions, the linear convergence of Picard's and approxi
mate Picard's methods, and the quadratic convergence of the quasilineariza
tion and approximate quasilinearization. These inequalities are further utilized
in Sections 1.7 and 1.8 to study right focal point boundary value problems for
the integro-differential and delay-differential equations, respectively.
Section 1.9 addresses necessary and sufficient conditions for the right dis
focality of linear differential equations. Here Markov, Descartes and Fekete
systems of solutions of linear differential equations for the right disfocality are
also introduced and their equivalence is shown. In Section 1.10 we present
so me easier tests for the right and eventual disfocalities of linear differential
equations. In Section 1.11 we construct the Green's function for linear differ
ential equations satisfying two-point (k, n - k) right focal boundary conditions.
We also determine the sign of each derivative of the Green's function.
In Section 1.12 first we introduce the concepts of upper and lower solutions
of (1.1.1), and then employ the results of Section 1.11 to accomplish the mono
tone convergence of the Picard iterative sehe me for the two-point (k, n - k)